3.8 Reflection principle
Some of this material is in the chapter of [Kunen] called “Easy consistency proofs”.
Let \phi (x_1, \ldots , x_ n) be a formula of set theory. Let us use the convention that this notation implies that all the free variables in \phi occur among x_1, \ldots , x_ n. Let M be a set. The formula \phi ^ M(x_1, \ldots , x_ n) is the formula obtained from \phi (x_1, \ldots , x_ n) by replacing all the \forall x and \exists x by \forall x\in M and \exists x\in M, respectively. So the formula \phi (x_1, x_2) = \exists x (x\in x_1 \wedge x\in x_2) is turned into \phi ^ M(x_1, x_2) = \exists x \in M (x\in x_1 \wedge x\in x_2). The formula \phi ^ M is called the relativization of \phi to M.
Theorem 3.8.1. Suppose given \phi _1(x_1, \ldots , x_ n), \ldots , \phi _ m(x_1, \ldots , x_ n) a finite collection of formulas of set theory. Let M_0 be a set. There exists a set M such that M_0 \subset M and \forall x_1, \ldots , x_ n \in M, we have
\forall i = 1, \ldots , m, \ \phi _ i^{M}(x_1, \ldots , x_ n) \Leftrightarrow \forall i = 1, \ldots , m, \ \phi _ i(x_1, \ldots , x_ n).
In fact we may take M = V_\alpha for some limit ordinal \alpha .
Proof.
See [Theorem 12.14, Jech] or [Theorem 7.4, Kunen].
\square
We view this theorem as saying the following: Given any x_1, \ldots , x_ n \in M the formulas hold with the bound variables ranging through all sets if and only if they hold for the bound variables ranging through elements of V_\alpha . This theorem is a meta-theorem because it deals with the formulas of set theory directly. It actually says that given the finite list of formulas \phi _1, \ldots , \phi _ m with at most free variables x_1, \ldots , x_ n the sentence
\begin{matrix} \forall M_0\ \exists M, \ M_0 \subset M\ \forall x_1, \ldots , x_ n \in M
\\ \phi _1(x_1, \ldots , x_ n) \wedge \ldots \wedge \phi _ m(x_1, \ldots , x_ n) \leftrightarrow \phi _1^ M(x_1, \ldots , x_ n) \wedge \ldots \wedge \phi _ m^ M(x_1, \ldots , x_ n)
\end{matrix}
is provable in ZFC. In other words, whenever we actually write down a finite list of formulas \phi _ i, we get a theorem.
It is somewhat hard to use this theorem in “ordinary mathematics” since the meaning of the formulas \phi _ i^ M(x_1, \ldots , x_ n) is not so clear! Instead, we will use the idea of the proof of the reflection principle to prove the existence results we need directly.
Comments (4)
Comment #7741 by Jake on
Comment #7744 by Laurent Moret-Bailly on
Comment #7988 by Stacks Project on
Comment #10045 by Einstein Wall Art on